University of Washington Math 523A Lecture 2 Martingales: concentration I Summary: University of Washington Math 523A Lecture 2 Martingales: concentration I Lecturer: Eyal Lubetzky April 3, 2009 The Hoeffding-Azuma concentration inequality Theorem 1 (Hoeffding `63). Let (Xt) be a martingale with respect to (Ft), and let c1, c2, . . . be real numbers such that |Xt - Xt-1| ct for all t. Then for any a > 0, P(|Xn - X0| a) 2 exp - a2 2 n i=1 c2 i . Proof. We will show a slightly stronger version of the above theorem: Instead of requiring that (Xt) is a martingale, we will assume that E[Xt | Xt-1] = Xt-1 for all t . (0.1) Let Yt = Xt - Xt-1. By the above assumptions E[Yt | Xt-1] = 0 and |Yt| ct for all t. The following simple claim says that, if Z is a r.v. bounded by 1 and with mean 0, then the expected value of exp(Z) is maximized when splitting the mass equally between ±1. Claim 2. Let Z be a random variable satisfying |Z| 1 and EZ = 0. Then Collections: Computer Technologies and Information Sciences